Elphick's graph energy lower-bound conjecture in terms of irregularity

Let $G$ be a connected graph with $n$ vertices, $m$ edges, eigenvalues $\lambda_1,\ldots,\lambda_n$ of its adjacency matrix, graph energy $E=\sum_{i=1}^n|\lambda_i|$, and vertex degrees $d_1,\ldots,d_n$. Define

$$\epsilon=\frac{n\sum_{i\sim j}\sqrt{d_i d_j}}{2m^2}.$$

Elphick's conjecture. The graph energy satisfies

$$E\geq\frac{n}{\epsilon}.$$

This conjecture was communicated by Clive Elphick and would improve the known lower bound involving the spectral radius, since $\beta=\lambda_1n/(2m)$ and $\beta\geq\epsilon\geq1$. Its resolution is not specified in the source.